WADS 2017: Algorithms and Data Structures pp 229-240

# Approximating Small Balanced Vertex Separators in Almost Linear Time

• Sebastian Brandt
• Roger Wattenhofer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10389)

## Abstract

For a graph G with n vertices and m edges, we give a randomized Las Vegas algorithm that approximates a small balanced vertex separator of G in almost linear time. More precisely, we show the following, for any $$\frac{2}{3}\le \alpha <1$$ and any $$0<\varepsilon <1-\alpha$$: If G contains an $$\alpha$$-separator of size K, then our algorithm finds an $$(\alpha +\varepsilon )$$-separator of size $$\mathcal O(\varepsilon ^{-1}K^2\log ^{1+o(1)} n)$$ in time $$\mathcal O(\varepsilon ^{-1}K^3m\log ^{2+o(1)} n)$$ w.h.p. In particular, if $$K\in \mathcal O({\text {polylog}}\, n)$$, then we obtain an $$(\alpha +\varepsilon )$$-separator of size $$\mathcal O(\varepsilon ^{-1}{\text {polylog}}\, n)$$ in time $$\mathcal O(\varepsilon ^{-1}m\,{\text {polylog}}\, n)$$ w.h.p. The presented algorithm does not require knowledge of K.

Due to space restrictions, no proofs are included in this version of the paper; the full version with a lot of additional material can be found at http://disco.ethz.ch/publications/wads2017-vertexsep.pdf.

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